Due to the fact that convex functions have a unique

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Due to the fact that convex functions have a unique global minimum, convexity plays an important role in optimization. For example, in numerical optimization convexity assures a global minimum to the problem. It is therefore important to first establish the convexity property when solving optimization problems. The following characterization of convexity applies to the solution spaces in such problems. Further ways of establishing convexity are discussed in (Boyd & Vandenberghe, Chaps. 2&3). If a function ݃ ሺ࢞ሻ is convex, then the set ݃ ሺ࢞ሻ ൑ ݁ is convex. Further, if functions ݃ ሺ࢞ሻǡ ݅ ൌ ͳǡ ǥ ǡ ݉ǡ are convex, then the set ሼ࢞ǣ ݃ ሺ࢞ሻ ൑ ݁ ǡ ݅ ൌ ͳǡ ǥ ǡ ݉ሽ is convex. In general, finite intersection of convex sets (that include hyperplanes and halfspaces) is convex. For general optimization problems involving inequality constraints: ݃ ሺ࢞ሻ ൑ ݁ ǡ ݅ ൌ ͳǡ ǥ ǡ ݉ ³ DQG ݈ , and equality constraints: ݄ ሺ࢞ሻ ൌ ܾ ǡ ݆ ൌ ͳǡ ǥ ǡ ݈ǡ W the feasible region for the problem is defined by the set: ܵ ൌ ሼ࢞ǣ ݃ ሺ࢞ሻ ൑ ݁ ǡ ݄ ሺ࢞ሻ ൌ ܾ ² The feasible region is a convex set if the functions: ݃ ݅ ൌ ͳǡ ǥ ǡ ݉ǡ are convex and the functions: ݄ ǡ ݆ ൌ ͳǡ ǥ ǡ ݈ǡ are linear. Note that these convexity conditions are sufficient but not necessary. Visit us and find out why we are the best! Master’s Open Day: 22 February 2014 Join the best at the Maastricht University School of Business and Economics! Top master’s programmes • 33 rd place Financial Times worldwide ranking: MSc International Business • 1 st place: MSc International Business • 1 st place: MSc Financial Economics • 2 nd place: MSc Management of Learning • 2 nd place: MSc Economics • 2 nd place: MSc Econometrics and Operations Research • 2 nd place: MSc Global Supply Chain Management and Change Sources: Keuzegids Master ranking 2013; Elsevier ‘Beste Studies’ ranking 2012; Financial Times Global Masters in Management ranking 2012 Maastricht University is the best specialist university in the Netherlands (Elsevier)
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Download free eBooks at Fundamental Engineering Optimization Methods 26 Mathematical Preliminaries 2.6 Vector and Matrix Norms Norms provide a measure for the size of a vector or matrix, similar to the notion of absolute value in the case of real numbers. A norm of a vector or matrix is a real-valued function with the following properties: 1. ԡ࢞ԡ ൒ Ͳ for all כ 2. ԡ࢞ԡ ൌ Ͳ if and only if ࢞ ൌ ૙ 3. ԡߙ࢞ԡ ൌ ȁߙȁԡ࢞ԡ for all ߙ א Թ 4. ԡ࢞ ൅ ࢟ԡ ൑ ԡ࢞ԡ ൅ ԡ࢟ԡ Matrix norms additionally satisfy: 5. || AB ||≤|| A || || B || Vector Norms. Vector p-norms are defined by ԡ࢞ԡ ൌ ሺσ ȁݔ ȁ ௜ୀଵ ǡ ݌ ൒ ͳ ² They include the 1-norm ԡ࢞ԡ ൌ σ ȁݔ ȁ ௜ୀଵ ³ the Euclidean norm P ԡ࢞ԡ ൌ ඥσ ȁݔ ȁ ௜ୀଵ ³
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